Topological Vector SpacesThe present book is intended to be a systematic text on topological vector spaces and presupposes familiarity with the elements of general topology and linear algebra. The author has found it unnecessary to rederive these results, since they are equally basic for many other areas of mathematics, and every beginning graduate student is likely to have made their acquaintance. Similarly, the elementary facts on Hilbert and Banach spaces are widely known and are not discussed in detail in this book, which is mainly addressed to those readers who have attained and wish to get beyond the introductory level. The book has its origin in courses given by the author at Washington State University, the University of Michigan, and the University of Tiibingen in the years 1958-1963. At that time there existed no reasonably complete text on topological vector spaces in English, and there seemed to be a genuine need for a book on this subject. This situation changed in 1963 with the appearance of the book by Kelley, Namioka et al. [1] which, through its many elegant proofs, has had some influence on the final draft of this manuscript. Yet the two books appear to be sufficiently different in spirit and subject matter to justify the publication of this manuscript; in particular, the present book includes a discussion of topological tensor products, nuclear spaces, ordered topological vector spaces, and an appendix on positive operators. |
Contents
Prerequisites | 1 |
Linear Algebra | 9 |
LOCALLY CONVEX TOPOLOGICAL | 36 |
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Common terms and phrases
adjoint algebraic assertion Banach lattice Banach space barreled barreled space bilinear bornological bounded sets bounded subsets called canonical map canonical order Chapter circled hull circled O-neighborhood circled subset closed subspace closure compact space compact subset contains continuous linear form continuous linear map convergence convex hull Corollary countable defined denote dense direct sum duality E₁ elements equicontinuous equicontinuous subsets equivalent Exercise exists F)-space F₁ family of bounded filter finite subset follows functions G₁ Hausdorff t.v.s. hence hyperplane implies inductive limit isomorphism L₁ lemma let F locally convex space locally convex topology Math metrizable neighborhood base non-empty normed space nuclear spaces order complete ordered vector space positive cone positive linear form precompact projective topology quasi-complete quotient reflexive respectively S-topology S₁ Section semi-norms semi-reflexive spectral strong dual subspace summable T₁ theorem vector lattice weakly
References to this book
Nonnegative Matrices in the Mathematical Sciences Abraham Berman,Robert J. Plemmons No preview available - 1994 |